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← JEE Main 2022 · 26 Jun · Shift 2
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JEE Main 2022 · 26 Jun · Shift 2, Mathematics question 4

lim⁡x→0cos⁡(sin⁡x)−cos⁡xx4\displaystyle\lim_{x\to0}\frac{\cos(\sin x)-\cos x}{x^4} is equal to :

Choose one option
Answer and worked solution

Official answer: C

Sum-to-product. cos⁡(sin⁡x)−cos⁡x=2sin⁡x+sin⁡x2sin⁡x−sin⁡x2\cos(\sin x)-\cos x=2\sin\dfrac{x+\sin x}2\sin\dfrac{x-\sin x}2.

Expand. x+sin⁡x2≈x\dfrac{x+\sin x}2\approx x and x−sin⁡x2≈x312\dfrac{x-\sin x}2\approx\dfrac{x^3}{12}, so the numerator ≈2⋅x⋅x312=x46\approx2\cdot x\cdot\dfrac{x^3}{12}=\dfrac{x^4}6.

Limit 16\dfrac16.

(C)

Topics:LimitsTrigonometric ratios and identities
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